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Dependent Types in Scala
Yao Li@lastland0
–-- Brent A. Yorgey, et al. Giving Haskell a Promotion.
Static type systems are the world’s most successful application of formal methods. Types are simple enough to make sense to programmers; they are tractable enough to be machine-checked on every compilation; they carry no run-time overhead; and they pluck a harvest of low-hanging fruit.
Types Are Specifications
sealed abstract class List[+A] {
def apply(n: Int): A
}
Types Are Specifications
• null vs. None
• Tony Hoare’s “billion-dollar mistake”
• Option types enforce programmers to check the “nullness” of a value in their implementation
Types Are Simplescala> val l = List(1, 2, 3)
l: List[Int] = List(1, 2, 3)
scala> l(3)
java.lang.IndexOutOfBoundsException: 3
at scala.collection.LinearSeqOptimized.apply(LinearSeqOptimized.scala:63)
at scala.collection.LinearSeqOptimized.apply$(LinearSeqOptimized.scala:61)
at scala.collection.immutable.List.apply(List.scala:86)
... 30 elided
Types Are Simple• They can not specify a red-black tree.
• Invariants of a red-black tree:
• Each node is either red or black.
• The root is black.
• All leaves are black.
• If a node is red, then both its children are black.
• Every path from a given node to any of its descendant leaves contains the same number of black nodes.
Dependent TypesFixpoint rep (A: Type) (n: nat) (a: A):
Vector A n:=
match n with
| O => VNil A
| S n’ => VCons A n’ a (rep A n’ a)
end.
Roadmap• In this talk, we will implement the following in Scala:
• a vector whose length information is in its type,
• a rep function which takes a number, and returns a Vector of exactly that length,
• an app function that takes two vectors, and returns a list whose length is the sum of their lengths,
• and an indexing method with compile-time bounds checking.
Demo
There’s more!
Dependent Types• Theorem prover.
• Propositions as types!
• Formal verification.
• For further reading: Verified Functional Algorithms, by Andrew Appel.
• Certified softwares.
• CompCert, VST, CertiKOS, FSCQ, Kami, etc.
Further Reading• Full dependent type languages:
• Gallina (Coq)
• Software Foundations, by Benjamin C. Pierce
• Idris
• Type-Driven Development with Idris, by Edwin Brady
• Agda
• …
Further Reading• Dependent Types in Haskell
• Dependently Typed Programming with Singletons, by Richard Eisenberg and Stephanie Weirich
• Depending on Types, by Stephanie Weirich (https://www.youtube.com/watch?v=n-b1PYbRUOY)
• The Influence of Dependent Types, by Stephanie Weirich (https://www.youtube.com/watch?v=GgD0KUxMaQs)
References• All the codes I have shown are written by myself, but I would not
know how to write them without the help of following materials:
• Dave Gurnell’s The Type Astronaut’s Guide to Shapeless.
• The source code of shapeless library: https://github.com/milessabin/shapeless/
• Miles Sabin’s demo at StrangeLoop 2013: https://github.com/milessabin/strangeloop-2013
• Miles Sabin’s dependently typed red-black tree: https://github.com/milessabin/tls-philly-rbtree-2016
